One-terminal fault location system that corrects for fault resistance effects

ABSTRACT

A one-terminal process for locating a fault associated with a multi-phase electric power transmission system is disclosed. The process is based on the principle that the impedance in a fault can be determined by correcting errors due to the interaction of fault resistance and load current. The fault may be a phase-to-ground fault or a multiple-phase fault.

FIELD OF THE INVENTION

The present invention relates generally to the field of protectiverelaying. More particularly, the invention relates to a system forautomatically locating faults on an electrical transmission line byprocessing voltage and current phasors measured at a single end of theline.

BACKGROUND OF THE INVENTION

Protective relaying generally involves the performance of one or more ofthe following functions in connection with a protected power or energysystem: (a) monitoring, which involves determining whether the system isin a normal or abnormal state; (b) metering, which involves measuringcertain electrical quantities; (c) protection, which typically involvestripping a circuit breaker in response to the detection of ashort-circuit condition; and (d) alarming, which provides a warning ofsome impending problem. Fault location is associated with the protectionfunction. It involves measuring critical system parameters and, when afault occurs, making an estimate of the fault location so that thefaulted line can be returned to service as quickly as possible.

The phasor diagrams in FIGS. 1A-1E illustrate the effect of faults onthe system voltages and currents. The diagrams are for effectivelygrounded systems, wherein the neutral is solidly grounded, and for theideal case of a zero fault resistance (R_(F) =0). However, they areillustrative of the effects of faults on other types of systems, e.g.,ungrounded and impedance grounded systems. In the diagrams, the dotted,uncollapsed voltage triangle exists in the source (the generator) andthe maximum collapse is at the fault location. The voltages between thesource and fault will vary between these extremes. The diagrams depictthe effects of various types of faults on the currents and voltages(represented by phasors) in the system. FIG. 1A depicts the phasors fornormal, balanced conditions; FIG. 1B depicts the phasors for athree-phase fault (V_(ab) =V_(bc) =V_(ca) =0 at the fault); FIG. 1Cdepicts the phasors for a phase b-to-phase c fault (V_(bc) =0 at thefault); FIG. 1D depicts the phasors for a phase b-to-phase c-to-groundfault (V_(bc) =V_(bg) =V_(cg) =0 at the fault); and FIG. 1E depicts thephasors for a phase a-to-ground fault (V_(ag) =0 at the fault).

An accurate estimate of the fault location is important to theutilities, particularly in bad weather and/or rough terrain, to avoidcumbersome searches and delays in line restoration. Accuracy isparticularly important for long lines because with long lines a largepercentage error in the fault location estimate represents aconsiderable distance. Furthermore, the fault condition may betemporary, due to fault clearing and/or a change in weather conditions,and not readily visible. In successful reclosing, accurate faultlocation information may be necessary to locate weak spots on the lineand to speed the analysis of the disturbance.

Fault location systems may be classified as two-terminal data systems orone-terminal data systems. With two-terminal data systems, voltages andcurrents are measured at opposite ends of the protected line(s). Thesesystems typically are more accurate than one-terminal data systems.However, two-terminal systems have a disadvantage in that communicationbetween the respective terminals is required. Since end-to-endcommunication is not always available and can be interrupted, therequirement for data from two ends of the protected line represents adisadvantage of two-terminal data systems. With one-terminal datasystems, only local voltages and currents are required. End-to-endcommunication is not required.

In one known one-terminal data system, certain initial values, both forthe argument difference and the fault distance, are assumed, and thecurrent and voltage at the fault point are determined. If these twoquantities are not in phase, new values of the argument difference andthe fault distance are assumed. This procedure is repeated until thecalculated fault current and the fault voltage are in phase. The lastcalculated value of the fault distance is assumed to be the correctvalue. However, small changes in the assumed value of the argumentdifference result in great changes of the calculated fault distance.Therefore, this system in many cases provides completely incorrectvalues or fails to converge toward a definite fault distance.

Another known system for locating faults with respect to a singlemonitoring point examines the time taken for a disturbance to travelfrom the monitoring point to the fault and back to the monitoring pointafter reflection at the fault point. A problem which could arise withthis system is that the reflected disturbance could be confused withother disturbances arriving at the monitoring point as a result ofreflections from other points in the transmission system. This couldresult in the protected section of the system being unnecessarilyremoved from service, when the fault is outside the protected section.

U.S. Pat. No. 4,559,491, Dec. 17, 1985, "Method and Device for Locatinga Fault Point On a Three-Phase Power Transmission Line," discloses amethod whereby currents and voltages are measured at a measuring pointarranged at one end of a section of a three-phase transmission line.FIG. 2 is a one-line schematic diagram of the disclosed system. Thetransmission line section under consideration has a length DL betweenits end points A and B. A fault locator FL is arranged adjacent to theend point A and is connected to the line via voltage and currenttransformers 1, 2 that feed measuring signals u and i to the faultlocator. The signals u and i are proportional to the voltages andcurrents at the point A. The line section has an impedance Z_(L). Afault of arbitrary type is assumed to have occurred at a point F at thedistance DF from the end point A. If n=DF/DL, the line impedance betweenthe points A and F is n×Z_(L) and between the points F and B the lineimpedance is (1-n)×Z_(L). The network located "behind" end point A has asource voltage E_(A) and an impedance Z_(A). The network located "aheadof" the end point B has a source voltage E_(B) and an impedance Z_(B).It is assumed that Z_(L) is a known parameter. The patent discloses thatZ_(A) may be known or may be calculated from measured values of currentsand voltages taken at the end point A before and after a fault, and thatZ_(B) may be known but, if not, should be determinable with anacceptable degree of accuracy so that its value can be set in the faultlocator FL.

When a fault occurs, the fault locator estimates the unknown distance DF(or the ratio n which gives the relative distance) from measured valuesof currents and voltages at the end point A before and after the faultand from pre-set or calculated values of the parameters Z_(A), Z_(B),Z_(L). To estimate the fault location, the system determines the faulttype and the measured currents and voltages are filtered for formationof their fundamental frequency components. Guided by the fault type andthe complex values of the fundamental frequency components of themeasured values, the impedance of the line section and the pre-set orcalculated values of the impedances of the networks lying ahead of andbehind the fault distance (n) are determined as the solution of thequadratic equation

    n.sup.2 +B×n+C=0,

where n is the fault distance and B and C are dependent on theimpedances and the fundamental frequency components of the measuredvalues. (In the below description of the present invention, the faultlocation parameter is referred to as "m"). A shortcoming of thistechnique is that values of source impedance Z_(A) and Z_(B) are neededif the error introduced by fault resistance is to be fully compensated.(In the description of the present invention, the source impedancesZ_(A), Z_(B) are referred to as Z_(s) and Z_(R)). Source impedanceschange due to the changes in network configuration and information abouttheir values is not readily available. A change in network configurationwill degrade the accuracy of this technique.

In application Ser. No. 08/515,274 filed by Novasel ("Novasel") entitled"One-Terminal Data Fault Location System," which is assigned to theassignee of the present invention, a system for locating faults fromone-terminal data is disclosed. Using a phase "a" fault as an example,Novasel determines the fault location based on the following equation:

    VR.sub.a =IR.sub.a mZ.sub.11 +I.sub.a2 D

where: VR_(a) is the voltage as measured from the relay;

IR_(a) is the compensated relay current;

Z₁₁ is the line positive sequence impedance;

I_(a2) is the negative sequence current;

R_(f) is equal to the fault resistance;

D is the fault resistance divided by the positive sequence distributionfactor; and

m is indicative of the fault location.

IR_(a) is further defined as I_(a) +3KI₀, and D is further defined as:##EQU1## where R_(f) is the fault resistance and K₁ is the positivesequence distribution factor.

The equation has two unknowns m and D. To solve for m and find the faultlocation, the voltage equation is split into real and imaginaryequations as follows:

    Re{VRx}=mRe{IR.sub.x Z.sub.11 }+DRe{I.sub.x2 }

and

    Im{VRx}=mIm{IR.sub.x Z.sub.11 }+DIm{I.sub.x2 }

With two equations and two unknowns, the equation can then be solved form. The equation accurately determines m for most fault conditions.However, under certain unique fault conditions, the equations couldbecome ill-conditioned. That is, the two equations become equivalent orcontradictory. Under such conditions, the resulting value for m will beinvalid.

Accordingly, there is a need for an improved one-terminal data faultlocation system that offers advantages over the prior art. The presentinvention provides such a system.

SUMMARY OF THE INVENTION

The present invention provides a fault location system for accuratelylocating a phase-to-ground fault or phase-to-phase fault associated withone or more conductors of an electric power transmission or distributionsystem. The system disclosed herein is useful in automaticallyestimating the location of faults in transmission lines.

The inventive process comprises measuring at least one voltage phasorand at least one current phasor. Each voltage phasor is indicative of anamplitude and phase associated with a voltage waveform at a firstprescribed location and each current phasor is indicative of anamplitude and phase associated with a current waveform at the firstprescribed location. A fault location parameter m indicative of thelocation of the fault is computed as: ##EQU2## where:

X_(x) =an imaginary portion of a faulted circuit impedance;

R=a real portion of a faulted circuit impedance;

X₁₁ =a positive sequence line reactance;

R₁₁ =a positive sequence line resistance;

I_(x) ' varies depending upon the fault type. In asingle-phase-to-ground fault, I_(x) ' is the current phasor pluscompensation for cross coupling effects; whereas it is a differencebetween current phasors in a multiple-phase fault; and

I_(m) also varies depending upon the fault type. In asingle-phase-to-ground fault, I_(m) is equal to a negative sequencecurrent of the current phasor;

whereas in a multiple-phase fault, I_(m) is a difference betweenpre-fault load currents and the current phasors.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A-1E are phasor diagrams depicting the effects of various typesof faults on the currents and voltages of a typical power system.

FIG. 2 is a schematic diagram referred to above in explaining one priorart fault location system.

FIG. 3 is a schematic diagram of a fault location system in accordancewith the present invention.

FIG. 3A is a single-line diagram of a transmission line system with afault through an impedance R_(f) on a homogeneous line of impedance Z₁between buses A and B.

FIG. 4 is a flowchart of an embodiment of a fault location processes inaccordance with the present invention.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

The present invention provides a system (methods and apparatus) forestimating a fault location parameter that is indicative of the locationof the fault. In preferred embodiments of the invention, voltage andcurrent phasors from one terminal of a protected line are required.Although the invention is described throughout the description inreference to a protective relay, it is not necessary that these phasorsbe provided by a protective relay per se, as they could be computed in aseparate processor. For example, the invention may be incorporated intoa separate fault location device. The invention may be implemented aspart of a fault-location software package for use with a protectiverelaying system.

I. System Description

FIG. 3 schematically depicts an exemplary microprocessor basedprotective relay in a three-phase power system wherein the presentinvention may be employed. Power source 12 provides voltage and currentto the three-phases a, b and c of the system. Current transformers 16provide a mechanism to supply a measure of the current flowing througheach respective phase of the power system to microprocessor based relay10. Similarly, voltage transformers 18 provide a mechanism to supply ameasure of the voltage on each respective phase of the power system tomicroprocessor based relay 10. Additionally, binary inputs may besupplied to the microprocessor based relay. Inside the relay, a tripdecision is made based on information obtained from the inputs. If adecision is made to trip, sufficient data is gathered from the lines todetermine the fault location. Thereafter, a signal is provided fromrelay 10 to the appropriate circuit breaker or circuit breakers 20,causing the circuit to open.

Phase currents and voltages are continuously monitored for faultconditions. The processes for determining when a fault has occurred arenumerous and complex, and are known to those skilled in the art.Accordingly, the description of such processes are left out of thepresent discussion for clarity and brevity. However, after a fault hasbeen detected, the fault location process executes according to thedetails set forth below.

FIG. 3A is a single-line diagram of the system, with a fault through animpedance R_(f) on a homogeneous line (impedance Z₁) between buses A andB. A single-phase representation is initially used as a preliminary stepin deriving the fault location process for a multi-phase transmissionsystem. The elements of the model represent physical quantities for thephase "x", where x is one of a, b or c. The elements of the model aredefined as follows:

V_(x) : Local bus voltage;

I_(x) : Current through the transmission line before the fault pointfrom the source E_(A) ;

R_(f) : Fault resistance (assumed to be real);

I_(f) : Current through the fault resistance;

V_(F) : Voltage across the fault resistance;

Z₁ : Impedance of the total transmission line;

I_(x-f) : Current flowing to the load after the fault point or thecontribution of current from the source E_(B) ;

m: Relay to fault distance as a ratio of the total line length.

The value m is representative of the fault location. A further step isrequired to translate m into a location distance in miles or kilometers.For example, if the length of the line between the buses A and B isknown, the distance to the fault can be found by multiplying the lengthof the line by the value m. Additional examples of converting m intomiles or kilometers are provided below.

For a single phase to ground fault (e.g., phase "a" to ground), a firstvariation of the process of the invention is employed. For multiplephase faults (e.g., phase "a" to phase "b" to ground), then a secondvariation of the process is employed to solve for m. The two processvariations are described below in reference to the flow chart of FIG. 4.

The present invention is best understood by referring to FIG. 4 inconjunction with the fault model of FIG. 3A. Before the process ofdetermining the fault location can begin, several constant values mustbe provided. For example, values for line positive sequence impedance,Z₁₁, and zero sequence impedance, Z₁₀, must be set into themicroprocessor based relay 10 (step 100). These constant values are usedto derive a cross coupling factor, K, a positive sequence reactance, X₁₁and the positive sequence resistance, R₁₁ (step 110).

In a three-phase system, the cross-coupling from other lines inducescurrent in the other phases. Therefore, the current in any one phasemust be compensated for the current induced in that phase by the otherphases. A "K factor" is used to compensate for the cross.. coupling. TheK factor is defined as: ##EQU3## This K factor is then maintained as aconstant in relay 10.

The relay then continuously monitors the lines for faults by measuringthe voltages and currents of all three phases (i.e., V_(a), V_(b),V_(c), I_(a), I_(b), I_(c)) (step 120). Each voltage and current valueconsists of a set of sampled values, for example; eight samples percycle. If no fault is detected, the set of sampled values are saved asthe load values. Thereafter, when a fault occurs there are two sets ofvoltages and currents available, the previously saved load values andpresent post fault values (130). As noted above, the method fordetermining whether a fault has occurred is beyond the scope of thepresent invention and therefore its description is left out for brevityand clarity. However, once a fault is detected the fault location isdetermined (step 140); otherwise the monitoring continues.

The fault type is determined next (i.e., single phase or multiple phase)according to conventional techniques and the process branchesaccordingly (step 150). As with the fault detection, fault typedetermination is well-known to those skilled in the art. Accordingly,details of fault type determination are left out of the presentdescription for clarity and brevity.

For a single phase to ground fault, the faulted phase voltages andcurrents are used to derive the sequence currents and voltages accordingto well-known sequence network techniques (step 160). Using a phase "a"fault as an example, I_(a) is converted to I_(a1), I_(a2) and I_(a0) andV_(a) is converted to V_(a1), V_(a2) and V_(a0). Next, the phase currentI_(x) is compensated for cross-coupling using the K factor (step 165).The compensated current in the phase that experiences the fault isdetermined as follows:

    I.sub.x '=I.sub.x +K*I.sub.x0                              (2)

where x is the faulted phase a, b or c.

After the current is compensated, the next step is to determine theimpedance, Z_(x), in the faulted phase. From the compensated phasecurrent, I_(x) ', and measured voltage, V_(x), the impedance, Z_(x), isdetermined according to equation (3) as described below (step 170).##EQU4## The reactance X_(x) and the resistance R_(x) are then derivedfrom the impedance value (step 170).

Finally, the difference in phase, α_(x), between the compensated currentand the negative sequence current is determined (step 180):

    α.sub.x =arg(I.sub.x ')-arg(I.sub.x2)                (4)

All of the values are then available to determine m (step 190) accordingto the equation: ##EQU5## As can be seen from the equation (5) above, ifthe tan α_(x) is equal to zero then the equation for m becomes: ##EQU6##indicating that there is no interaction of fault resistance and loadcurrent. Additionally, if ##EQU7## then, the denominator in equation (5)is equal to zero. So, m is calculated as in equation (5.1).

From m, the fault distance in miles (or kilometers) can be determinedaccording to the equation: ##EQU8## where X_(unit) is the per unit(e.g., per mile or per kilometer) reactance of the transmission line.

The process for determining fault location in a multiple-phase faulttype is somewhat different than that employed in a single-phase fault.In a multiple-phase fault, adjustments are made to the equation before mcan be determined. In particular, the values for V_(x) and I_(x) areredefined to account for the multiple phases. Additionally, ΔI_(x)replaces the negative sequence current in the α equation. The values forV_(x), I_(x) ', and ΔI_(x) are derived according to Table 1 shown below(step 162).

                  TABLE 1                                                         ______________________________________                                        Substitutions for multiple phase faults.                                      Fault Type                                                                             V.sub.x     I.sub.x '   .increment.I.sub.x                           ______________________________________                                        a-b or a-b-G                                                                           V.sub.ab = V.sub.a - V.sub.b                                                              I.sub.ab ' = I.sub.a - I.sub.b                                                            I.sub.ab ' - (I.sub.Lda - I.sub.Ldb)         b-c or b-c-G                                                                           V.sub.bc = V.sub.b - V.sub.c                                                              I.sub.bc ' = I.sub.b - I.sub.c                                                            I.sub.bc ' - (I.sub.Ldb - I.sub.Ldc)         c-a or c-a-G                                                                           V.sub.ca = V.sub.c - V.sub.a                                                              I.sub.ca ' = I.sub.c - I.sub.a                                                            I.sub.ab ' - (I.sub.Ldc - I.sub.Lda)         a-b-c or any of the above                                                                          any of the above                                                                          any of the above                             a-b-c-G                                                                       ______________________________________                                    

Those derived values are used to determine the impedance Z_(x),according to equation (3), the same equation used in the single phasefault location calculation. From the impedance, Z_(x), reactance, X_(x),and resistance, R_(x), are determined (step 170). Next, α is derivedaccording to the equation below:

    α=arg(I.sub.x ')-arg(ΔI.sub.x)                 (7)

Thereafter, all the values necessary to calculate m are available (step190) and m can be determined using the same equation (5) above.Thereafter, the fault location can be determined as in a single phasefault (step 200).

II. Derivations and Additional Details

As is described below, m is derived from the mathematical equivalence ofthe circuit of FIG. 3A.

    V.sub.x =I.sub.x 'mZ.sub.11 +I.sub.f R.sub.f               (8)

(The equation assumes that the fault impedance is real.) From thisequation the impedance as measured from Terminal A is derived. The valueof that impedance (i.e., Z_(x)) is equal to the impedance of the line upto the point of the fault (i.e., mZ₁) plus the impedance after the fault(i.e., the impedance through the fault in combination with the impedancethrough the load). The impedance measured at terminal A can berepresented as: ##EQU9## The line current is represented as I_(x) 'because, in a three-phase system, the cross-coupling from other linesinduces current in the other phases. Therefore, the current in any onephase must be compensated for the current induced in that phase by theother phases. A K factor is used for compensating for the crosscoupling. The K factor is determined by: ##EQU10## where: Z₁₀ is thezero sequence line impedance;

Z₁₁ is the positive sequence line impedance.

Thus, the compensated current (also referred to as the relay current) inthe phase that experiences the fault can be determined as follows:

    I.sub.x '=I.sub.x +K*I.sub.x0                              (10)

where I_(x0) is the zero sequence phase current.

The line current, I_(x), can be further represented as the currentcontributed by the source 12 (E_(A)) through the fault resistance,I_(Af), plus the load current, I_(1dx). Accordingly, I_(x) ' can befurther represented as:

    I.sub.x '=I.sub.Af I.sub.Ldx +K*I.sub.x0                   (11)

Notice that the compensated current only includes the current throughthe fault contributed by the source E_(A) and does not include currentcontributed from the source E_(B). From equation (2), it can also beappreciated that if R_(f) is zero, i.e., the fault is a complete shortcircuit, then:

    Z.sub.x =mZ.sub.1                                          (12)

where Z₁ is total positive sequence impedance of the transmission line.

However, if the fault is not assumed to be a short circuit, then thereis some impedance difference which must be accounted for in theequation. In such a case:

    Z.sub.x =mZ.sub.1 +ΔZ                                (13)

where ΔZ represents the difference between the actual fault impedanceand a short circuit fault impedance.

Accordingly, by correcting the equation for the value of ΔZ, a moreaccurate Z_(x) will be obtained. To that end, referring back to equation(2) it can be seen that: ##EQU11##

The fault current, I_(f), is not readily determinable from the relay. Assuch, a substitution is needed. For a single phase to ground faultwherein a serial connection of positive, negative, and zero sequencenetworks for the faulted system exists, the following equations aresatisfied: ##EQU12## Although the above equations are satisfied, thefault sequence currents are also unmeasurable directly. Accordingly, avalue measurable from the relay must be substituted for the faultcurrents. The fault sequence currents are proportional to the linesequence currents. However, other current contributors may be in thesystem, contributing to the fault current. For example, in FIG. 3A,generator 14 may also contribute some current to the fault. Thus, thefault sequence currents are a function of both current sources 12, 14.Accordingly, the fault sequence currents can be represented as:##EQU13## where K₁ is the ratio of positive sequence current from source12 to that of source 14. ##EQU14## where K₂ is the ratio of negativesequence current from source 12 to that of source 14. ##EQU15## where K₀is the ratio of zero sequence current from source 12 to that of source14.

As seen from the equation (15) above, there are essentially four optionsfor replacing the fault current with a current measurable from therelay; the first option substitutes positive sequence current for faultcurrent as shown below: ##EQU16## Since I₁₁ is the positive sequencecurrent of the fault component of line current at the relay, loadcurrent should be excluded. This means that the load current must bemeasured and saved before fault inception so that it can be removed fromthe line current. For light loads, the load current is more affected byline charging current. As a result, it is difficult to compensate forthe charging current exactly as it is distributed along the line.

The second option substitutes negative sequence current for faultcurrent as shown below: ##EQU17## Assuming that the power system isunder normal conditions and thus balanced, I_(x2) is independent of loadcurrent.

The third option substitutes zero sequence current for fault current asshown below: ##EQU18## Like negative sequence current, there is no needto know load current when using zero sequence current.

The fourth option substitutes positive and negative sequence currentsfor the fault current as shown below: ##EQU19## Assuming that K₁ isequivalent to K₂. The fourth option can be written as: ##EQU20##Starting from the equation for phase current, the fourth option can bewritten in terms of load and zero sequence currents:

    I.sub.x =I.sub.Ldx +I.sub.x1 +I.sub.x2 +I.sub.x0           (24)

then,

    I.sub.x1 +I.sub.x2 =I.sub.x +I.sub.Ldx +I.sub.x0           (25)

As a result, ##EQU21## The fourth option can thus be expressed so thatno positive or negative sequence currents are required; however, loadand zero sequence currents are required. Moreover, this option dependson the assumption that K₁ and K₂ are equivalent and real.

Considering the four options, the second option, which employs negativesequence current, involves less computations and does not need pre-faultinformation, assuming K₂ is real. By contrast, the third option, whichemploys zero sequence current, is the simplest approach, if K₀ canaccurately be assumed to be real. However, K₀ may have a relativelylarge angle because the zero sequence network depends on the groundinggrids at substations and the earth along the transmission line. Althoughthe description accompanying the FIG. 3A above uses negative sequencecurrent, any of the above options can be substituted.

After evaluating the fault current substitutions, the impedance equationfor the circuit of FIG. 3A can b written in terms of negative sequencecurrent as follows: ##EQU22## If R_(f) and K₂ are real, then only I_(X2)and I_(x) ' have a phase angle. Moreover, the relationship betweenI_(x2) and I_(x) ' can be written as: ##EQU23## where:

    α.sub.x =arg(I.sub.x ')-arg(I.sub.x2)                (4)

Using equations (30) and (4), the impedance equation can be expressed asfollows:

    Z.sub.x =mZ.sub.11 +ye.sup.-jα.sbsp.x                (29)

where: ##EQU24## Splitting equation (29) into real and imaginary partsyields:

    R.sub.x =mR.sub.11 +y cos α.sub.x                    (31)

and,

    X.sub.x mX.sub.11 -y sin α.sub.x                     (32)

Rearranging equations (31) and (32) yields:

    R.sub.x -mR.sub.11 =y cos α.sub.x                    (33)

and

    X.sub.x -mX.sub.11 =-y sin α.sub.x                   (34)

Dividing the equations (34) by equation (33) yields: ##EQU25## The valuem can be isolated by rewriting equation (35) as shown below: ##EQU26##Significantly, the variable y is factored out of the equations above.Thus, as long as the phases are correct, the magnitude of the currents(e.g. I_(x2) and I_(x)) become irrelevant to the fault locationdeterminations.

The fault location formula derived above is for single phase to groundfaults. However, the same principle can be applied to multiple phasefaults substituting the original voltage equation as indicated belowwith a new voltage equation: ##EQU27## where R_(p) is the resistancethrough the multiple phase fault and V_(x), I_(x) and ΔI_(x) are definedin Table 1 above. The values for V_(x) and I_(x) ' can be substitutedinto the single-phase to ground fault equations above to derive themultiple phase fault distance.

In addition to the difference noted above, the determination of α issomewhat different for multiple-phase faults. In multiple phase faults,ΔI_(x) is substituted for the negative sequence current used in thesingle phase fault α calculation. Thus α is rewritten as:

    α=arg(I.sub.x ')-arg(ΔI.sub.x)                 (7)

With these two adjustments, m can be determined in multi-phase faultsjust as with single phase faults, i.e., according to equation (5) shownabove.

The above description of preferred embodiments is not intended toimpliedly limit the scope of protection of the following claims. Thus,for example, except where they are expressly so limited, the followingclaims are not limited to applications involving three-phase powersystems. Moreover, the claims are not limited to fault location systemsassociated with any particular section (i.e., transformer, feeder, highpower transmission line, etc.) of a power distribution system.

I claim:
 1. A system for locating a fault associated with one or moreconductors of a multiple phase electric power transmission ordistribution system, said fault being one of the following group offault types: phase-to-ground, phase-to-phase-to-ground, phase-to-phase,and phase-to-phase-to-phase, comprising:measuring means for measuring atleast one voltage phasor and at least one current phasor, each saidvoltage phasor being indicative of an amplitude and phase associatedwith a voltage waveform at a first prescribed location and each saidcurrent phasor being indicative of an amplitude and phase associatedwith a current waveform at said first prescribed location; processingmeans coupled to said measuring means for computing a fault locationparameter m indicative of the location of the fault, wherein m iscomputed as: ##EQU28## where: X_(x) =an imaginary portion of a faultedcircuit impedance; R=a real portion of a faulted circuit impedance; X₁₁=a positive sequence line reactance; R₁₁ =a positive sequence lineresistance; I_(x) '=said current phasor compensated for cross couplingeffects in a single-phase to ground fault and a difference betweencurrent phasors in a multiple-phase fault; and I_(m) =a negativesequence current of said current phasor in a single-phase-to-groundfault and a difference between pre-fault load currents and said currentphasors in a multiple-phase fault.
 2. A system as recited in claim 1,wherein said faulted circuit impedance is computed as: ##EQU29## where:V_(x) =said voltage phasor in a single-phase fault and a differencebetween voltage phasors in a multiple-phase fault; and,I_(x) '=saidcurrent phasor compensated for cross-coupling effects in a single phasefault and a difference between current phasors in a multiple-phasefault.
 3. A system as recited in claim 1, further comprising processingmeans for determining a location L in distance units to the faultaccording to the following equation: ##EQU30## wherein X₁₁ is a measuredpositive sequence line reactance; and, X_(unit) is a per unit linereactance.
 4. A system according to claim 3, wherein the per unit linereactance is in miles.
 5. A system according to claim 3, wherein the perunit line reactance is in kilometers.
 6. A system as recited in claim 1,wherein said current phasor in a single-phase fault is compensated byadding a compensation current via processing means, wherein saidcompensation current being equal to: ##EQU31## where Z₁₁ is a positivesequence line impedance, Z₁₀ is a zero sequence line impedance andI_(X0) is the zero sequence current for the faulted phase.
 7. A processfor locating a fault associated with one or more conductors of amultiple phase electric power transmission or distribution system, saidfault being one of the following group of fault types: phase-to-ground,phase-to-phase-to-ground, phase-to-phase, and phase-to-phase-to-phase,comprising the steps of:a) measuring at least one voltage phasor and atleast one current phasor, each said voltage phasor being indicative ofan amplitude and phase associated with a voltage waveform at a firstprescribed location and each said current phasor being indicative of anamplitude and phase associated with a current waveform at said firstprescribed location; b) computing a fault location parameter mindicative of the location of the fault, wherein m is computed as:##EQU32## where: X_(x) =an imaginary portion of a faulted circuitimpedance; R_(x) =a real portion of a faulted circuit impedance; X₁₁ =apositive sequence line reactance; R₁₁ =a positive sequence lineresistance; I_(x) '=said current phasor compensated for cross couplingeffects in a single-phase to ground fault and a difference betweencurrent phasors in a multiple-phase fault; and I_(m) =a negativesequence current of said current phasor in a single-phase-to-groundfault and a difference between pre-fault load currents and said currentphasors in a multiple-phase fault.
 8. A process as recited in claim 7,wherein said faulted circuit impedance is computed as: ##EQU33## where:V_(x) =said voltage phasor in a single-phase fault and a differencebetween voltage phasors in a multiple-phase fault; and,I_(x) '=saidcurrent phasor compensated for cross-coupling effects in a single phasefault and a difference between current phasors in a multiple-phasefault.
 9. A process as recited in claim 7, further comprising the stepof determining a location L in distance units to the fault according tothe following equation: ##EQU34## wherein X₁₁ is a measured positivesequence line reactance; and, X_(unit) is a per unit line reactance. 10.A process as recited in claim 9, wherein the per unit line reactance isin miles.
 11. A process as recited in claim 9, wherein the per unit linereactance is in kilometers.
 12. A process as recited in claim 7, whereinsaid step b) comprises the step of compensating the current phasor in asingle-phase fault by adding a compensation current, wherein saidcompensation current being computed according to the equation: ##EQU35##where Z₁₁ is a positive sequence line impedance, Z₁₀ is a zero sequenceline impedance and I_(x0) is the zero sequence current for the faultedphase.